Диссертация (1137342), страница 16
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Section 4.2 is devoted to the derivation of Fredholm determinant representation of the Jimbo-Miwa-Ueno isomonodromic tau function. It starts from a recast of the original rank N Fuchsian system with n regular704.1. Introductionsingular points on P1 in terms of a Riemann-Hilbert problem. In Subsection 4.2.2we associate to it, via a decomposition of n-punctured Riemann sphere into pairsof pants, n − 2 auxiliary Riemann-Hilbert problems of Fuchsian type having only 3regular singular points.
Section 4.2.3 introduces Plemelj operators acting on functions holomorphic on the annuli of the pants decomposition, and deals with theirbasic properties. The main result of the section is formulated in Theorem 4.11 ofSubsection 4.2.4, which relates the tau function of a Fuchsian system with prescribedgeneric monodromy to a Fredholm determinant whose blocks are expressed in termsof 3-point Plemelj operators.
In Subsection 4.2.5, we consider in more detail the example of n = 4 points and show that the Fredholm determinant representation canbe efficiently used for asymptotic analysis of the tau function. In particular, Theorem 4.13 provides a generalization of the Jimbo asymptotic formula for Painlevé VIvalid in any rank and up to any asymptotic order.In Section 4.3 we explain how the principal minor expansion of the Fredholm determinant leads to a combinatorial structure of the series representations for isomonodromic tau functions.
Theorem 4.15 of Subsection 4.3.1 shows that 3-point Plemeljoperators written in the Fourier basis are given by sums of a finite number of infinite Cauchy type matrices twisted by diagonal factors. Combinatorial labeling of theminors by N -tuples of charged Maya diagrams and partitions is described in Subsection 4.3.2.Section 4.4 deals with rank N = 2.
Hypergeometric representations of the appropriate 3-point Plemelj operators are listed in Lemma 4.28 of Subsection 4.4.1. Theorem 4.30 provides an explicit combinatorial series representation for the tau functionof the Garnier system. In the final subsection, we explain how Fredholm determinantof the Borodin-Olshanski hypergeometric kernel arises as a special case of our construction. Appendix contains a proof of a combinatorial identity expressing Nekrasovfunctions in terms of Maya diagrams instead of partitions.PerspectivesIn an effort to keep the chapter of reasonable length, we decided to defer the studyof several straightforward generalizations of our approach to separate publications.These extensions are outlined below together with a few more directions for futureresearch:1. In higher rank N > 2, it is an open problem to find integral/series representations for general solutions of 3-point Fuchsian systems and to obtain an explicitdescription of the Riemann-Hilbert map.
There is however an important exception of rigid systems having two generic regular singularities and one singularityof spectral multiplicity (N − 1, 1); these can be solved in terms of generalized hypergeometric functions of type N FN −1 . The spectral condition is exactly whatis needed to achieve factorization in Lemma 4.26. The results of Section 4.4can therefore be extended to Fuchsian systems with two generic singular pointsat 0 and ∞, and n−2 special ones. The corresponding isomonodromy equations(dubbed GN,n−3 system in [Tsu]) are the closest higher rank relatives of PainlevéVI and Garnier system.
It is natural to expect their tau functions to be relatedon the 2D CFT and gauge theory side, respectively, to WN conformal blocks714. Fredholm determinant and Nekrasov sum representations of isomonodromic tau functionswith semi-degenerate fields [FLitv12, Bul] and Nekrasov partition functions of4D linear quiver gauge theories with the gauge group U (N )⊗(n−3) .In the generic non-rigid case the 3-point solutions depend on (N − 1) (N − 2)accessory parameters and may be interpreted as matrix elements of a generalvertex operator for the WN algebra. They should also be related to the so-calledTN gauge theory without lagrangian description [BMPTY].2. Fredholm determinants and series expansions considered in the present work areassociated to linear pants decompositions of P1 \{n points}, which means thatevery pair of pants has at least one external boundary component (see Fig.
4.2a).Plemelj operators assigned to each trinion act on spaces of functions on internalboundary circles only. To be able to deal with arbitrary decompositions, inaddition to 4 operators a[k] , b[k] , c[k] , d[k] appearing in (4.20) one has to introduce5 more similar operators associated to other possible choices of ordered pairs ofboundary components.b)a)c)Figure 4.2: (a) Linear and (b) Sicilian pants decompositionof P1 \{6 points}; (c) gluing 1-punctured torus from a pairof pants.A (tri)fundamental example where this construction becomes important is knownin the gauge theory literature under the name of Sicilian quiver (Fig.
4.2b). Already for N = 2 the monodromies along the triple of internal cicles of this pantsdecomposition cannot be simultaneously reduced to the form “1+rank 1 matrix” by factoring out a suitable scalar piece. The analog of expansion (4.87) inTheorem 4.30 will therefore be more intricate yet explicitly computable. Sincethe identification [ILTe] of the tau function of the Garnier system with a Fouriertransform of c = 1 Virasoro conformal block does not put any constraint on theemployed pants decomposition, Sicilian expansion of the Garnier tau functionmay be used to produce an analog of Nekrasov representation for the corresponding conformal blocks.
It might be interesting to compare the results obtainedin this way against instanton counting [HKS].Extension of the procedure to higher genus requires introducing additional simple (diagonal in the Fourier basis) operators acting on some of the internalannuli. They give rise to a part of moduli of complex structure of the Riemannsurface and correspond to gluing a handle out of two boundary components.Fig. 4.2c shows how a 1-punctured torus may be obtained by gluing two boundary circles of a pair of pants. The gluing operator encodes the elliptic modulus,which plays a role of the time variable in the corresponding isomonodromic724.1. Introductionproblem.
Elliptic isomonodromic deformations have been studied e.g. in [K00],where the interested reader can find further references.3. It is natural to wonder to what extent the approach proposed in the presentwork may be followed in the presence of irregular singularities, in particular,for Painlevé I–V equations. The contours of appropriate isomonodromic RHPsbecome more complicated: in addition to circles of formal monodromy, theyinclude anti-Stokes rays, exponential jumps on which account for Stokes phenomenon [FIKN]. We will sketch here a partial answer in rank N = 2. Forthis it is useful to recall a geometric representation of the confluence diagramfor Painlevé equations recently proposed by Chekhov, Mazzocco and Rubtsov[CM, CMR], see Fig.
4.3. To each of the equations (or rather associated linearproblems) is assigned a Riemann surface with a number of cusped boundarycomponents. They are obtained from Painlevé VI 4-holed sphere using twosurgery operations: i) a “chewing-gum” move creating from two holes with kand l cusps one hole with k + l + 2 cusps and ii) a cusp removal reducing thenumber of cusps at one hole by 1. The cusps may be thought of as representingthe anti-Stokes rays of the Riemann-Hilbert contour.VdegIII(D8)VIII(D7)VIIII(D6 )II FNIVIII JMFigure 4.3: CMR confluence diagram for Painlevé equations.An extension of our approach is straightforward for equations from the upperpart of the CMR diagram and, more generally, when the Poincaré ranks ofall irregular singular points are either 12 or 1. The associated surfaces maybe decomposed into irregular pants of three types corresponding to solvableRHPs: Gauss hypergeometric, Whittaker and Bessel systems (Fig. 4.4).
Theyserve to construct local Riemann-Hilbert parametrices which in turn producethe relevant Plemelj operators.The study of higher Poincaré rank seems to require new ideas. Moreover, evenfor Painlevé V and Painlevé III Fredholm determinant expansions naturally giveseries representations of the corresponding tau functions of regular type, first734. Fredholm determinant and Nekrasov sum representations of isomonodromic tau functionsGaussWhittakerBesselFigure 4.4: Some solvable RHPs in rank N = 2: Gauss hypergeometric (3 regular punctures), Whittaker (1 regular + 1 of Poincarérank 1) and Bessel (1 regular + 1 of rank 21 ).proposed in [GIL13] and expressed in terms of irregular conformal blocks of[G, BMT, GT].
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